Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats


2    PARTICULAR CASES AND EXAMPLES       249

PROBLEMS

1.    Let G be a locally compact group, A a dense subset of G, ft a left Haar measure on G,
and H a /^-measurable subset of G with the following property: for each se A, the
sets sH n QH and H n (J^H are ^-negligible. Show that either H or its complement is
negligible (prove that the measure <pH " p is left-invariant).

2.    Let G be a locally compact group, ft a left Haar measure on G, and A, B subsets of G.
(a)   Suppose that one of the following two conditions is satisfied: (a) A is /i-integ-
rable; (ft) ju*(A) < + o, and B is /^-measurable. Show that in either case the function
/O) = fji*(sA n B) is uniformly continuous on G with respect to a right-invariant
distance on G. (For any two subsets M, N of G, put

p(M, N) = />((M n CN) u (N n CM)).

Consider first the case where A is compact, and show that for each e > 0 there exists
a neighborhood U of e in G such that p(sA n B, stA n B) <^ e for ail s e G and all
/ e U. Then apply Problem 5 of Section 13.9. If B is /^-measurable and /z*(A) < + oo,
observe that there exists a decreasing sequence (An) of jn-integrable subsets of G con-
taining A and such that inf(/x(An)) = ft-(A), and show that

B) = inf(/^*(5An n B))

(Section 13.9, Problem 2(a)). Use the fact that ju,(An - An+1) tends to 0 with 1/w.)

(b)    If A is /z-integrable and /x*(B) < + oo , then the function / is also uniformly
continuous with respect to a left-invariant distance on G. If also A"1 is /z-integrable,

then      f(s) <& = /x(A~I)/x*(B). (Reduce to the case where B is a-integrable, and
JG

observe that in that case /x(,yA n B) = p,(A n .s^B), and that <psA. n B = ^SA^B and
<psA(0 = <ptA-i(.s).)

(c)    Deduce from (a) that in the two cases considered there, the interiors of AB and BA
are nonempty if A and B are not ju-negligible.

(d)    In the group G = SL2(R), give an example of a compact set A and a jit-measurable
set B such that the function f(s)  p,(sA. r\ B) is not uniformly continuous with respect
to a left-invariant measure on G. (Observe that there exists a sequence (tn) of elements
of G tending to e and a sequence (sn) of elements of G such that the sequence s^1tnsn
tends to the point at infinity.)

3.    Let G be a locally compact group, p, a left Haar measure on G, and A an integrable
subset of G such that /i(A) > 0. Show that the set H(A) of elements s e G such that
/x(A) = /x(A n sA) is a compact group. (Use Problem 2 to show that H(A) is closed
in G. To show that H(A) is compact, consider a compact subset B of A such that
/i(B) > i/x(A), and prove that H(A) <= BB'1.)

4.    Let G be a commutative locally compact group, written additively. Let /x be a Haar
measure on G and let A, B be two integrable subsets of G.

(a)   For each 5 e G let

A' = <rf(A, B) = A u (B + s),       W  rs(A, B) = (A - ,y)nB.

Show that /z(A') + /4B') = /z(A) 4- MB) and that A' + B'^A + B.  (Note that

A + <f> = (/> for all subsets A of G.) onto L(X, JLC) for 1 <p g + oo.